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Mirrors > Home > MPE Home > Th. List > Mathboxes > imaiun1 | Structured version Visualization version GIF version |
Description: The image of an indexed union is the indexed union of the images. (Contributed by RP, 29-Jun-2020.) |
Ref | Expression |
---|---|
imaiun1 | ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexcom4 3246 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐵) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐵)) | |
2 | vex 3495 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 2 | elima3 5929 | . . . . 5 ⊢ (𝑦 ∈ (𝐵 “ 𝐶) ↔ ∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐵)) |
4 | 3 | rexbii 3244 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐵)) |
5 | eliun 4914 | . . . . . . 7 ⊢ (〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 〈𝑧, 𝑦〉 ∈ 𝐵) | |
6 | 5 | anbi2i 622 | . . . . . 6 ⊢ ((𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ (𝑧 ∈ 𝐶 ∧ ∃𝑥 ∈ 𝐴 〈𝑧, 𝑦〉 ∈ 𝐵)) |
7 | r19.42v 3347 | . . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐵) ↔ (𝑧 ∈ 𝐶 ∧ ∃𝑥 ∈ 𝐴 〈𝑧, 𝑦〉 ∈ 𝐵)) | |
8 | 6, 7 | bitr4i 279 | . . . . 5 ⊢ ((𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐵)) |
9 | 8 | exbii 1839 | . . . 4 ⊢ (∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐵)) |
10 | 1, 4, 9 | 3bitr4ri 305 | . . 3 ⊢ (∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 “ 𝐶)) |
11 | 2 | elima3 5929 | . . 3 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) ↔ ∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)) |
12 | eliun 4914 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 “ 𝐶)) | |
13 | 10, 11, 12 | 3bitr4i 304 | . 2 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶)) |
14 | 13 | eqriv 2815 | 1 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1528 ∃wex 1771 ∈ wcel 2105 ∃wrex 3136 〈cop 4563 ∪ ciun 4910 “ cima 5551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-iun 4912 df-br 5058 df-opab 5120 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 |
This theorem is referenced by: trclimalb2 39949 |
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